Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
552 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×#137 | C2 | semidirect |
| 3. C2×C5×C4⋊Q8 | C2 | semidirect |
| 4. C5×#497 | C2 | semidirect |
| 5. C5×#494 | C2 | semidirect |
| 6. C5×#492 | C2 | semidirect |
| 7. C4×C5×M16 | C2 | semidirect |
| 26. #3152 | C5 | direct (N × H) |
| 41. C5×M16 | D4 | semidirect |
| 50. C2×#137 | C2×C5 | semidirect |
| 52. C2×C4⋊Q8 | C2×C5 | semidirect |
| 53. #497 | C2×C5 | semidirect |
| 54. #494 | C2×C5 | semidirect |
| 55. #492 | C2×C5 | semidirect |
| 56. C4×M16 | C2×C5 | semidirect |
| 103. C4×C5 | #137 | semidirect |
| 119. M16 | C5×D4 | semidirect |
| 151. C5 | #3152 | direct (N × H) |
| 161. C4 | C5×#137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#137 | 2 |
| C5×#532 | 2 |
| C2×C5×C4⋊Q8 | 2 |
| C5×#497 | 2 |
| C5×#494 | 2 |
| C5×#492 | 2 |
| C4×C5×M16 | 2 |
| #3152 | 5 |